MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY NGUYEN HAI SON NO GAP OPTIMALITY CONDITIONS AND SOLUTION STABILITY FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY SEMILINEAR ELL[.]
MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY NGUYEN HAI SON NO-GAP OPTIMALITY CONDITIONS AND SOLUTION STABILITY FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY SEMILINEAR ELLIPTIC EQUATIONS DOCTORAL DISSERTATION OF MATHEMATICS Hanoi - 2019 MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY NGUYEN HAI SON NO-GAP OPTIMALITY CONDITIONS AND SOLUTION STABILITY FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY SEMILINEAR ELLIPTIC EQUATIONS Major: MATHEMATICS Code: 9460101 DOCTORAL DISSERTATION OF MATHEMATICS SUPERVISORS: Dr Nguyen Thi Toan Dr Bui Trong Kien Hanoi - 2019 COMMITTAL IN THE DISSERTATION I assure that my scientific results are new and righteous Before I published these results, there had been no such results in any scientific document I have responsibilities for my research results in the dissertation Hanoi, April 3rd , 2019 On behalf of Supervisors Author Dr Nguyen Thi Toan Nguyen Hai Son i ACKNOWLEDGEMENTS This dissertation has been carried out at the Department of Fundamental Mathematics, School of Applied Mathematics and Informatics, Hanoi University of Science and Technology It has been completed under the supervision of Dr Nguyen Thi Toan and Dr Bui Trong Kien First of all, I would like to express my deep gratitude to Dr Nguyen Thi Toan and Dr Bui Trong Kien for their careful, patient and effective supervision I am very lucky to have a chance to work with them, who are excellent researchers I would like to thank Prof Jen-Chih Yao for his support during the time I visited and studied at Department of Applied Mathematics, Sun Yat-Sen University, Kaohsiung, Taiwan (from April, 2015 to June, 2015 and from July, 2016 to September, 2016) I would like to express my gratitude to Prof Nguyen Dong Yen for his encouragement and many valuable comments I would also like to especially thank my friend, Dr Vu Huu Nhu for kind help and encouragement I would like to thank the Steering Committee of Hanoi University of Science and Technology (HUST), and School of Applied Mathematics and Informatics (SAMI) for their constant support and help I would like to thank all the members of SAMI for their encouragement and help I am so much indebted to my parents and my brother for their support I thank my wife for her love and encouragement This dissertation is a meaningful gift for them Hanoi, April 3rd , 2019 Nguyen Hai Son ii CONTENTS i ii iii COMMITTAL IN THE DISSERTATION ACKNOWLEDGEMENTS CONTENTS TABLE OF NOTATIONS INTRODUCTION Chapter 0.1 0.2 0.3 PRELIMINARIES AND AUXILIARY RESULTS Variational analysis 0.1.1 Set-valued maps 0.1.2 Tangent and normal cones Sobolev spaces and elliptic equations 13 0.2.1 Sobolev spaces 13 0.2.2 Semilinear elliptic equations 20 Conclusions 24 Chapter NO-GAP OPTIMALITY CONDITIONS FOR DISTRIBUTED CONTROL PROBLEMS 25 1.1 Second-order necessary optimality conditions 26 1.1.1 An abstract optimization problem 26 1.1.2 Second-order necessary optimality conditions for optimal control problem 27 1.2 Second-order sufficient optimality conditions 40 1.3 Conclusions 57 Chapter NO-GAP OPTIMALITY CONDITIONS FOR BOUNDARY CONTROL PROBLEMS 58 2.1 Abstract optimal control problems 59 2.2 Second-order necessary optimality conditions 66 2.3 Second-order sufficient optimality conditions 75 2.4 Conclusions 89 Chapter UPPER SEMICONTINUITY AND CONTINUITY OF THE SOLUTION MAP TO A PARAMETRIC BOUNDARY CONTROL PROBLEM 91 3.1 Assumptions and main result 92 3.2 Some auxiliary results 94 iii 3.2.1 Some properties of the admissible set 94 3.2.2 First-order necessary optimality conditions 98 3.3 Proof of the main result 100 3.4 Examples 104 3.5 Conclusions 109 GENERAL CONCLUSIONS LIST OF PUBLICATIONS 110 111 REFERENCES 112 iv TABLE OF NOTATIONS N := {1, 2, } R set of positive natural numbers |x| absolute value of x ∈ R Rn n-dimensional Euclidean vector space ∅ empty set x∈A x is in A x∈ /A x is not in A A ⊂ B(B ⊃ A) A is a subset of B A*B A is not a subset of B A∩B intersection of the sets A and B A∪B union of the sets A and B A\B set difference of A and B A×B Descartes product of the sets A and B [x1 , x2 ] the closed line segment between x1 and x2 kxk norm of a vector x kxkX norm of vector x in the space X X∗ topological dual of a normed space X X ∗∗ topological bi-dual of a normed space X hx∗ , xi canonical pairing hx, yi canonical inner product B(x, δ) open ball with centered at x and radius δ B(x, δ) closed ball with centered at x and radius δ BX open unit ball in a normed space X BX closed unit ball in a normed space X dist(x; Ω) distance from x to Ω {xk } sequence of vectors xk xk → x xk converges strongly to x (in norm topology) xk * x xk converges weakly to x ∀x for all x ∃x there exists x A := B A is defined by B f :X→Y function from X to Y f (x), Fr´echet derivative of f at x ∇f (x) f 00 (x), ∇2 f (x) set of real numbers Fr´echet second-order derivative of f at x Lx , ∇x L Fr´echet derivative of L in x Lxy , ∇2xy L Fr´echet second-order derivative of L in xand y ϕ : X → IR extended-real-valued function domϕ effective domain of ϕ epiϕ epigraph of ϕ suppϕ the support of ϕ F :X⇒Y multifunction from X to Y domF domain of F rgeF range of F gphF graph of F kerF kernel of F T (K, x) Bouligand tangent cone of the set K at x T [ (K, x) adjoint tangent cone of the set K at x T (K, x, d) second-order Bouligand tangent set of the set K at x in direction d T 2[ (K, x, d) second-order adjoint tangent set of the set K at x in direction d N (K, x) normal cone of the set K at x ∂Ω ¯ Ω boundary of the domain Ω Ω0 ⊂⊂ Ω Ω0 ⊂ Ω and Ω0 is compact Lp (Ω) the space of Lebesgue measurable functions f closure of the set Ω and R Ω |f (x)|p dx < +∞ L∞ (Ω) ¯ C(Ω) the space of bounded functions almost every Ω ¯ the space of continuous functions on Ω ¯ M(Ω) the space of finite regular Borel measures m,p m,p W (Ω), W0 (Ω), W s,r (Γ), Sobolev spaces H m (Ω), H0m (Ω) W −m,p (Ω)(p−1 + p0−1 = 1) the dual space of W0m,p (Ω) X ,→ Y X is continuous embedded in Y X ,→,→ Y X is compact embedded in Y i.e id est (that is) a.e almost every s.t subject to p page w.r.t with respect to The proof is complete INTRODUCTION Motivation Optimal control theory has many applications in economics, mechanics and other fields of science It has been systematically studied and strongly developed since the late 1950s, when two basic principles were made One was the Pontryagin Maximum Principle which provides necessary conditions to find optimal control functions The other was the Bellman Dynamic Programming Principle, a procedure that reduces the search for optimal control functions to finding the solutions of partial differential equations (the Hamilton-Jacobi equations) Up to now, optimal control theory has developed in many various research directions such as non-smooth optimal control, discrete optimal control, optimal control governed by ordinary differential equations (ODEs), optimal control governed by partial differential equations (PDEs), (see [1, 2, 3]) In the last decades, qualitative studies for optimal control problems governed by ODEs and PDEs have obtained many important results One of them is to give optimality conditions for optimal control problems For instance, J F Bonnans et al [4, 5, 6], studied optimality conditions for optimal control problems governed by ODEs, while J F Bonnans [7], E Casas et al [8, 9, 10, 11, 12, 13, 14, 15, 16, 17], C Meyer and F Trăoltzsch [18], B T Kien et al [19, 20, 21, 22], A Răosch and F Trăoltzsch [23, 24] derived optimality conditions for optimal control problems governed by elliptic equations It is known that if u¯ is a local minimum of F , where F : U → R is a differentiable functional and U is a Banach space, then F (¯ u) = This a first-order necessary optimality condition However, it is not a sufficient condition in case of F is not convex Therefore, we have to invoke other sufficient conditions and should study the second derivative (see [17]) Better understanding of second-order optimality conditions for optimal control problems governed by semilinear elliptic equations is an ongoing topic of research for several researchers This topic is great value in theory and in applications Second-order sufficient optimality conditions play an important role in the numerical analysis of nonlinear optimal control problems, and in analyzing the sequential quadratic programming algorithms (see [13, 16, 17]) and in studying the stability of optimal control (see [25, 26]) Second-order necessary optimality conditions not only provide criterion of finding out stationary points but also help us in constructing sufficient optimality conditions Let us briefly review some results on this topic For distributed control problems, i.e., the control only acts in the domain Ω in Rn , E Casas, T Bayen et al [11, 13, 16, 27] derived second-order necessary and sufficient optimality conditions for problem with pure control constraint, i.e., a(x) ≤ u(x) ≤ b(x) a.e x ∈ Ω, (1) and the appearance of state constraints More precisely, in [11] the authors gave second-order necessary and sufficient conditions for Neumann problems with constraint (1) and finitely many equalities and inequalities constraints of state variable y while the second-order sufficient optimality conditions are established for Dirichlet problems with constraint (1) and a pure state constraint in [13] T Bayen et al [27] derived second-order necessary and sufficient optimality conditions for Dirichlet problems in the sense of strong solution In particular, E Casas [16] established second-order sufficient optimality conditions for Dirichlet control problems and Neumann control problems with only constraint (1) when the objective function does not contain control variable u In [18], C Meyer and F Trăoltzsch derived second-order sufficient optimality conditions for Robin control problems with mixed constraint of the form a(x) ≤ λy(x) + u(x) ≤ b(x) a.e x ∈ Ω and finitely many equalities and inequalities constraints For boundary control problems, i.e., the control u only acts on the boundary , E Casas and F Trăoltzsch [10, 12] derived second-order necessary optimality conditions while the second-order sufficient optimality conditions were established by E Casas et al in [12, 13, 17] with pure pointwise constraints, i.e., a(x) ≤ u(x) ≤ b(x) a.e x A Răosch and F Trăoltzsch [23] gave the second-order sufficient optimality conditions for the problem with the mixed pointwise constraints which has unilateral linear form c(x) ≤ u(x) + γ(x)y(x) for a.e x ∈ Γ We emphasize that in above papers, a, b ∈ L∞ (Ω) or a, b ∈ L∞ (Γ) Therefore, the control u belongs to L∞ (Ω) or L∞ (Γ) This implies that corresponding Lagrange multipliers are measures rather than functions (see [19]) In order to avoid this disadvantage, B T Kien et al [19, 20, 21] recently established second-order necessary optimality conditions for distributed control of Dirichlet problems with mixed statecontrol constraints of the form a(x) ≤ g(x, y(x)) + u(x) ≤ b(x) a.e x ∈ Ω with a, b ∈ Lp (Ω), < p < ∞ and pure state constraints This motivates us to develop and study the following problems (OP 1) : Establish second-order necessary optimality conditions for Robin boundary control problems with mixed state-control constraints of the form a(x) ≤ g(x, y(x)) + u(x) ≤ b(x) a.e x ∈ Γ, where G(u) := g(·, ζ(u)) + λu By Φp := G−1 (K), we denote the admissible set of problem (1.13)–(1.14), i.e., Φp = {u ∈ Lp (Ω) | G(u) ∈ K} Note that ∇G(u) = gy (·, ζ(u))ζ (u) + λI, where I is the identity operator In the sequel, we shall show that, under assumptions (A1.2) and (A1.3), ∇G(¯ u) is a surjective operator This fact plays an important role in establishing optimality conditions Definition 1.1.10 The function u¯ ∈ Φp is said to be a locally optimal solution of problem (1.13)–(1.14) if there exists ε > such that f (u) ≥ f (¯ u) ∀u ∈ BU (¯ u, ) ∩ Φp Problem (1.13)–(1.14) is associated with the Lagrangian: L(u, e∗ ) = f (u) + he∗ , G(u)i = Z Z L(x, ζ(u), u)dx + Ω e∗ (g(·, ζ(u)) + λu)dx, Ω with e∗ ∈ Lp (Ω) Lemma 1.1.11 Suppose that assumptions (A1.1), (A1.2) and (A1.3) are satisfied Then, G is of class C and L(·, e∗ ) is second-order Fr´echet differentiable around u¯ and the following formulae are valid: ∇u L(¯ u, e∗ ) = φ¯ + Lu (¯ y , u¯) + λe∗ , 0 where φ¯ ∈ W 2,p (Ω) ∩ W01,p (Ω) is a unique solution of the so-called adjoint equation −∆φ¯ + h (·, y¯)φ¯ = L [·] + e∗ g [·] in Ω, y y y φ¯ = on Γ (1.15) and ∇2uu L(¯ u, e∗ )(u, u) Z ¯ yy [x]z dx, Lyy [x]zu2¯,u + 2Lyu [x]zu¯,u u + Luu [x]u2 + e∗ gyy [x]zu2¯,u − φh u ¯,u = Ω where zu¯,u is a solution of equation (1.11) corresponding to u := u¯ and v := u Proof For the proof, we consider the function ψ(t) = L(¯ u + tu, e∗ ) Z = L(x, ζ(¯ u + tu), u¯ + tu) + e∗ (g(x, ζ(¯ u + tu)) + λ(¯ u + tu) dx Ω Under (A1.1) and (A1.2), G is of class C and L(·, e∗ ) is second-order Fr´echet differentiable around u¯, and we have ∇u L(¯ u)u = ψ (0), ∇2uu L(¯ u)(u, u) = ψ 00 (0) 34 By a simple computation, we get Z Ly (x, ζ(¯ u + tu), u¯ + tu)ζ (¯ u + tu)u + Lu (x, ζ(¯ u + tu), u¯ + tu)u dx ψ (t) = ΩZ u + tu))ζ (¯ u + tu)u + λu dx e∗ gy (x, ζ(¯ + Ω Hence Z ψ (0) = ZΩ = ζ (¯ u)∗ (Ly [x] + e∗ gy [x])u + Lu [x]u + λe∗ u dx (φ¯ + Lu [x] + λe∗ )u(x)dx, Ω where φ¯ := ζ (¯ u)∗ (Ly [·] + e∗ gy [·]) = (A∗ )−1 (Ly [·] + e∗ gy [·]) and A is defined by (1.10) By definition of A∗ , φ¯ is a solution of the adjoint equation (1.15) Also, we have u)(u, u) = ψ 00 (0) ∇2uu L(¯ Z Lyy [x](ζ (¯ u)u)2 + 2Lyu [x](ζ (¯ u)u)u + Luu [x]u2 + Ly [x]ζ 00 (¯ u)u2 dx = Ω Z e∗ gyy [x](ζ (¯ u)u)2 + e∗ gy [x]ζ 00 (¯ u)u2 dx + Ω Z Lyy [x]zu2¯,u + 2Lyu [x]zu¯,u u + Luu [x]u2 + Ly [x]zu¯,uu dx = Ω Z e∗ gyy [x]zu2¯,u + e∗ gy [x]zu¯,uu dx + Ω Z Lyy [x]zu2¯,u + 2Lyu [x]zu¯,u u + Luu [x]u2 + e∗ gyy [x]zu2¯,u dx = Ω Z (Ly [x] + e∗ gy [x])zu¯,uu dx + Ω Z = Lyy [x]zu2¯,u + 2Lyu [x]zu¯,u u + Luu [x]u + e ∗ gyy [x]zu2¯,u Z ¯ u¯,uu dx (A∗ φ)z dx + Ω Ω By definition of A and zu¯,uu , we have ∇2uu L(¯ u)(u, u) Z = ZΩ = Lyy [x]zu2¯,u ∗ + 2Lyu [x]zu¯,u u + Luu [x]u + e gyy [x]zu2¯,u Z ¯ φA(z u ¯,uu )dx dx + Lyy [x]zu2¯,u + 2Lyu [x]zu¯,u u + Luu [x]u2 + e∗ gyy [x]zu2¯,u dx − Ω ZΩ ¯ yy [x]z dx φh u ¯,u Ω The proof is complete From Corollary 0.1.12, we have an explicit formula of T [ (K, u¯), which is proved in [61] (see also [44, Theorem 8.5.1,p 324] and [27, Lemma 4.11]) 35 Lemma 1.1.12 Let u¯ ∈ K Then, the following assertions are valid: (i) The tangent cone is given by T [ (K, u¯) = {v ∈ Lp (Ω) | v(x) ∈ T [ ([a(x), b(x)], u¯(x)), a.e x ∈ Ω} ≥ if u¯(x) = a(x) p = v ∈ L (Ω) | v(x) ≤ if u¯(x) = b(x) n o a.e x ∈ Ω (ii) The normal cone is given by N (K, u¯) = {u∗ ∈ Lp (Ω) | u∗ (x) ∈ N ([a(x), b(x)], u¯(x)) a.e x ∈ Ω} (iii) For all u∗ ∈ N (K, u¯), one has T [ (K, u¯) ∩ (u∗ )⊥ = {v ∈ T [ (K, u¯) | u∗ (x)v(x) = a.e x ∈ Ω}, where (u∗ )⊥ := {u ∈ Lp (Ω) | R Ω u∗ (x)u(x)dx = 0} (iv) K is polyhedric at any z¯ ∈ K Lemma 1.1.13 Suppose that assumptions (A1.2), (A1.3) are valid and u¯ ∈ Φp Then, problem (1.13)–(1.14) satisfies Robinson’s constraint qualification at u¯ Proof We shall show that (1.5) is valid at u¯, i.e., Lp (Ω) = ∇G(¯ u)U − K(G(¯ u)) Here ∇G(¯ u)u = gy [·]zu¯,u + λu Taking any v ∈ Lp (Ω), we consider equation −∆z + λhy [x]+gy [x] ≥0 λ 1,p 2,p W (Ω) ∩ W0 (Ω) (λhy [·] + gy [·])z v = λ λ in Ω, z|Γ = By (A1.3), a.e x ∈ Ω Therefore, the equation has a unique solution z ∈ Putting u = −∆z + hy [·]z, we see that u ∈ Lp (Ω) and v = gy [·]zu¯,u + λu with zu¯,u := z Hence we have v = ∇G(¯ u)u − (G(¯ u) − G(¯ u)) This implies that u¯ is regular The proof is complete Lemma 1.1.14 Suppose that assumptions (A1.2) and (A1.3) are valid and u¯ ∈ Φp Then, problem (1.13)–(1.14) satisfies the strongly extended polyhedricity condition at u¯ Proof We first notice that under assumption (A1.2), G is of class C around u¯ Here, we have ∇G(¯ u) = ∇y g(·, y¯)ζ (¯ u) + λI, where I is the identity mapping of Lp (Ω) Under assumption (A1.3), the proof of Lemma 1.1.13 implies that for any v ∈ Lp (Ω), we can find u ∈ Lp (Ω) such that 36 v = ∇G(¯ u)u Hence, ∇G(¯ u) is surjective By (iv) of Lemma 1.1.12, K is polyhedric at G(¯ u) Then the conclusion of the lemma follows from [47, Proposition 3.54] The proof is complete Given u¯ ∈ Φp , the cone of all critical directions of problem (1.13)–(1.14) is defined by Cp (¯ u) := d ∈ U | ∇f (¯ u)d ≤ 0, ∇G(¯ u)d ∈ T [ (K, G(¯ u)) Recall that Z ∇f (¯ u)d = (Ly [x]zu¯,d + Lu [x]d)dx and ∇G(¯ u)d = gy [·]zu¯,d + λd Ω From Lemma 1.1.12, we can rewrite ≥ if x ∈ Ω a p Cp (¯ u) = d ∈ L (Ω) | (Ly [x]zu¯,d + Lu [x]d)dx ≤ 0, ∇G(¯ u)d(x) ≤ if x ∈ Ωb Ω Z o a.e , where Ωa := {x ∈ Ω | G(¯ u)(x) = a(x)}, Ωb := {x ∈ Ω | G(¯ u)(x) = b(x)} (1.16) The following theorem provides second-order necessary optimality conditions for problem (1.13)–(1.14) Theorem 1.1.15 Suppose that assumptions (A1.1)–(A1.3) are satisfied and u¯ is a locally optimal solution of (1.13)–(1.14) There exist a unique e∗ ∈ Lp (Ω) and a 0 unique φ¯ ∈ W 2,p (Ω) ∩ W01,p (Ω) such that the following conditions are valid: (i) The adjoint equation: −∆φ¯ + h [·]φ¯ = L [·] + e∗ g [·] in Ω, y y y φ¯ = on Γ; (1.17) (ii) The stationary conditions in u: ∇u L(¯ u, e∗ ) = φ¯ + Lu [·] + λe∗ = 0, e∗ ∈ N (K, G(¯ u)); (1.18) (iii) The non-negative second-order condition: ∇2uu L(¯ u, e∗ )(v, v) ≥ ∀v ∈ Cp (¯ u), where ∇2uu L(¯ u, e∗ )(v, v) Z = Lyy [x]zu2¯,v + 2Lyu [x]zu¯,v v + Luu [x]v + e∗ gyy [x]zu2¯,v Ω ¯ yy [x]z dx − φh u ¯,v 37 Proof From Lemmas 1.1.13 and 1.1.14, we see that, all assumptions of Lemma 1.1.3 are fulfilled According to Lemma 1.1.3, for each v ∈ Cp (¯ u), there exists a multiplier e∗ ∈ Lq (Ω) such that ∇u L(¯ u, e∗ ) = 0, e∗ ∈ N (K, G(¯ u)) (1.19) ∇2uu L(¯ u, e∗ )(v, v) ≥ (1.20) and From the first relation of (1.19) and Lemma 1.1.11, there exists φ¯ ∈ W 2,q (Ω) ∩ W01,q (Ω) such that φ¯ + Lu [·] + λe∗ = and φ¯ is the unique solution of (1.17) Hence we get (i) and (ii) Finally, from (1.20) and Lemma 1.1.11, we obtain assertion (iii) We next show that the multipliers e∗ and φ¯ are unique and independent of critical directions v In fact, assume that for v1 ∈ Cp (¯ u), there exist multipliers e∗ and φ¯1 satisfying (1.17) and (1.18) Let us put e∗0 := e∗ − e∗1 and φ0 := φ¯ − φ¯1 We then have −∆φ0 + hy [·]φ0 = e∗0 gy [·] in Ω, φ0 = on Γ and φ0 + λe∗0 = It follows that φ0 is a solution of the equation −∆φ0 + (λhy [·] + gy [·]) φ0 = in Ω, φ0 = on Γ λ By (A1.3), this equation has a unique solution φ0 = and so e∗0 = Hence e∗ = e∗1 and φ¯ = φ¯1 Consequently, e∗ and φ are unique and independent of v The proof is complete The following example illustrates how to use necessary conditions to find stationary points In this example, we use first-order necessary optimality conditions to show that (0; 0) is a stationary point However, it does not satisfy second-order necessary optimality conditions Example 1.1.16 Let Ω be the open unit ball in R2 with boundary Γ and p = We consider the following problem of finding u ∈ L4 (Ω) and y ∈ W 2,4 (Ω) ∩ W01,4 (Ω) which minimize the objective function Z F (y, u) = Ω s.t y(x) + u0 (x)y (x) dx − 3 − ∆y + y = u in Ω, Z (u(x) − u0 (x)) dx + Ω Z u4 (x)dx Ω y|Γ = 0, − ≤ u ≤ a.e x ∈ Ω, where u0 (x) = u0 (x1 , x2 ) := 41 (1 − x21 − x22 ) and ρ > Suppose that (¯ y , u¯) is a locally optimal solution We have 1 L(x, y, u) = (y + u0 (x)y ) − (u − u0 (x))2 + u4 , g(x, y) = 0, λ = 38 h(x, y) = y , We define Ωa := {x ∈ Ω | u¯(x) = a} and u(x) := 1 if x ∈ Ω−1 , otherwise a.e −1 if x ∈ Ω1 , Let zu¯,u be the unique solution of the following equation −∆zu¯,u + 3¯ y zu¯,u = u in Ω, zu¯,u = on Γ (1.21) Since K = {v ∈ L4 (Ω) | −1 ≤ v(x) ≤ a.e.}, we get v ∈ T [ (K, u¯) ⇔ v(x) ∈ T [ ([−1, 1], u¯(x)) a.e Hence we have u ∈ T [ (K, u¯) Multiplying (1.21) by u0 and integrating by part, we get Z (−zu¯,u ∆u0 + 3¯ y zu¯,u u0 − uu0 )dx = 0, Ω or equivalently, Z (zu¯,u + 3¯ y zu¯,u u0 − uu0 )dx = Ω It follows that Z Z (Ly [x]zu¯,u + Lu [x]u)dx = Ω u − u0 + 2¯ u3 u dx + 3¯ y u0 zu¯,u + −¯ ZΩ = Z (1 + 2¯ u2 )¯ uudx (1 + 2¯ u )udx = Ω1 ∪Ω−1 ZΩ (−1)dx ≤ = Ω1 ∪Ω−1 Hence we obtain u ∈ C4 (¯ u) By Theorem 1.1.15, there exist functions e∗ ∈ L4/3 (Ω) and φ¯ ∈ W 2,4/3 (Ω) ∩ 1,4/3 W0 (Ω) such that the following conditions hold: (i) The adjoint equation: −∆φ¯ + 3¯ y φ = (1 + 3u0 y¯2 ) in Ω, φ¯ = on Γ; (ii) The stationarity condition with respect to u: φ¯ − u¯ − u0 + 2¯ u3 + e∗ = a.e.; e∗ ∈ N (K, u¯) From (ii), we get e∗ (x) ≥ if x ∈ Ω1 , ≤ if x ∈ Ω−1 , = otherwise 39 a.e and he∗ , ui = On the other hand, we have ∗ Z Z ∗ he , ui = e∗ (x)u(x)dx e (x)u(x)dx = Ω1 ∪Ω−1 Ω = if e∗ = a.e., ∗ ∗ =− e (x)dx + e (x)dx < otherwise Ω1 Ω−1 Z Z It follows that e∗ = Since u0 = 14 (1 − x21 − x22 ) and Ω is the open unit ball in R2 , the adjoint equation has a unique solution φ¯ = u0 From this and e∗ = and the first relation of (ii), we get −¯ u + 2¯ u3 = Hence u¯ = a.e or u¯ = ± √12 a.e If u¯ = a.e x ∈ Ω then y¯ = Now we will show that (¯ u, y¯) = (0; 0) is not satisfied second-order necessary optimality conditions In fact, we have Lyy = 6u0 y, Luu = −1 + 2u2 , Lyu = 0, and hyy = 6y, and so ∇2uu L(¯ u, e∗ )(v, v) Z −v dx for all v ∈ C4 (¯ u) = (1.22) Ω On the other hand, it follows from definition of critical cone that n Z o C4 (¯ u) = v ∈ L (Ω) | (zu¯,v − u0 v)dx ≤ , Ω where zu¯,v solves the following equation −∆y = v in Ω, zu¯,u = on Γ It is easily seen that if v = then zu¯,v = u0 This implies that ∈ C4 (¯ u) From (1.22) we get ∇2uu L(¯ u, e∗ )(1, 1) Z −1dx = −|Ω| < = Ω which is not impossible Therefore, the second-order necessary optimality conditions is invalid and so (0, 0) is not an optimal solution 1.2 Second-order sufficient optimality conditions To derive second-order sufficient optimality conditions for elliptic optimal control problems we usually use two different norms: a stronger one (for instance L∞ ) for differentiability and neighborhoods and a weaker one (for instance L2 ) for the coercivity (see, for instance, [3] and [13]) However, by this approach there will be a gap between the second-order necessary conditions and the second-order sufficient conditions, i.e., we can not obtain a common critical cone under which both the second-order necessary conditions and the second-order sufficient conditions are valid In this section, instead of using the two-norm method we exploit the structure of the objective function in 40 order to derive a common critical cone to the problem for the case p = 2, N ∈ {2, 3} and the objective function has the form L(x, y, u) = ϕ(x, y) + α(x)u + β(x)u2 , (1.23) where ϕ : Ω × R → R is a Carath´eodory function and α, β ∈ L∞ (Ω) In the sequel, we need the following assumptions (A1.1)0 Function ϕ : Ω × R → R is a Carath´eodory function of class C with respect to the second variable, ϕ(x, 0) ∈ L1 (Ω) and for each M > there are a constant Kϕ,M > and a function ϕM ∈ L2 (Ω) such that ∂ϕ (x, y) ≤ ϕM (x), ∂y